Compressive behavior and buckling response of

Carbon Nanotubes

Dianyun Zhang∗, Aswath Rangarajan†, Anthony M. Waas‡∗

College of Engineering, University of Michigan, Ann Arbor, MI-48109, USA

Carbon nanotubes (CNTs) are allotropes of Carbon, which are supposed to possessvery high strength. There has been relatively less analysis on the buckling response andthe compressive behavior of the Carbon Nanotubes. The current study deals with thecompressive behavior and buckling response of CNTs, while dealing with it as a springsystem as well as by performing lattice calculations using GULP (General Utility LatticeProgram) and Finite Element Analysis (FEA) on the Carbon Nanotube structure, whileapproximating them to be structures made of beams. The spring model shows responseswhich are very similar to the buckling response for long nanotubes, as seen in the literature.

I. Introduction

Carbon Nanotubes (CNTs) are allotropes of Carbon, which are supposed to possess very high strength.The possible applications to which such high strength materials can be used in vary from simple items suchas tennis rackets, going right to next generation armors. The stability and buckling of CNTs has beenwidely discussed, but most of the theories do not conform to the experimental results accurately.

There have been a few experimental studies on the axial stability of multi walled CNTs. The work doneby Waters et al. on the compression of CNTs having very low aspect ratios shows the CNTs to have anincreasing equilibrium load after the initial buckling, which occurs around 2.5 µN .

From the work of Akita et al[1], it can be seen that for CNTs having very large aspect ratios, the forcedisplacement curves have a linear slope, and after a decrease in the slope at the buckling load, the linearrelationship between the load and displacement is maintained. This behavior of the long CNTs can be com-pared with spring systems which have been designed to have the same force-displacement characteristics.The results also show that carbon nanotube bundles seem to exhibit super elastic behavior, making themsuitable candidates for the next generation armors.

Hirooka et al[3] have worked on the loading and unloading of CNTs having intermediate aspect ratios(between 40 and 60). These results show that the CNTs come back to their initial states on unloading evenafter the buckling load has been reached. However, one of the CNTs having a length of 800 mm, showsforce-displacement behavior which is similar to the behavior observed by Akita et al[1].

TEM images of buckled configurations of CNTs show ripple like distortions on the CNT . It can also beseen that one side is in tension while the other side is in compression during the buckling of the CNT , andhence, the load bearing capacity of the CNT even upon buckling.

Non Local Continuum theories can be used to create a 1D model of the CNT , but that would not capturethe exact buckling mechanism of the CNT , since the effects due to the shell type of structure of the nanotubeis ignored in this model.

∗Graduate Student, Department of Aerospace Engineering†Graduate Student, Department of Aerospace Engineering‡Professor, Department of Aerospace Engineering and Department of Mechanical Engineering

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II. Models used for analyzing CNTs

A. Spring Model to characterize the CNT :

1. Two DOF Model

Two Degree of freedom spring systems have been found to have buckling characteristics which are similar tothat of long CNTs. Thus, in the analysis which follows, a non linear spring model is used in order to obtainthe buckling characteristics of the CNT .

2. Description of the model

The model consists of spring systems which are used in the place of the CNT . Considering the CNT as ashell, during buckling, one side of the CNT is in tension while the other side is in compression. Hence, therewill be an additional load bearing capacity due to the tensile side of the CNT . This idea is implementedusing linear and rotational springs as follows. The side which is in compression is characterized by a combi-nation of linear translational springs and non linear rotational springs, while the side which is in tension ischaracterized by non linear translational springs. Also, the linear springs do not bend, i.e. they are rigid inbending. The equilibrium curves are obtained by the minimization of the Potential energy.

π =kt1R

2

4+2kr1θ]

2−4kr2θ]4−P (L−(L−R)Cosθ)+

1

2kt2(L−(L−R)Cosθ)2+

1

4kt3(L−(L−R)Cosθ)4 (1)

Figure 1. Two DOF Spring Model

3. Parametrization

The spring stiffnesses are parameterized in order to obtain a correlation to the experimental data. Theparametrization is done as follows.

NON DIMENSIONAL PARAMETERS

p =P

4kr1(2)

r =R

L(3)

p =P

4kr1(4)

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k1 =kt1L

2

32kr1(5)

k2 =kt2L

2

32kr1(6)

k3 =kt3L

2

32kr1(7)

k22 =kr2kr1

(8)

π0 =1

2θ2 − k22θ

4 + 2k1r2 + 4k2(1 − (1 − r)Cosθ)2 + p(1 − r)Cosθ − p+ 2k3(1 − (1 − r)Cosθ)4 (9)

B. GULP

1. Brief overview of GULP:

General Utility Lattice Program (GULP) is used for performing lattice energy calculations. In the presentcase, the program is used to obtain the minimum energy configuration of the CNT . The process is iterative,ie., the minimum energy configuration for a particular length of the CNT is obtained using the previousleast energy configuration of the CNT as a base. GULP helps in obtaining the configuration of the CNTwhich minimizes the lattice energy of the atoms of the CNT .

Figure 2. CNT used in the Simulation

2. Simulations for various CNT configurations

In this study, armchair configurations of CNTs were analyzed using GULP. A detailed description of thevarious configurations of the CNT is given in referenceThe coordinates of the CNT are obtained usingTubeGen, given in the reference In this study, Single Walled CNTs (SWCNT) are studied.

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3. Potential used for the Simulation

In this study, the Tersoff Potential is used for creating the interactions between the atoms of the CNT . TheTersoff potential has been found to be useful for simulating the behavior of molecules which have covalentbonding between the atoms[]. The Tersoff potential is a bond order potential and takes into account thevarious interactions due to the torsion and stretching behavior between the various carbon atoms. It takesinto account Multi Body potentials and hence the non local behavior of the CNT , which is more realisticthan a local model for describing the behavior of the CNT .

A detailed description of the Tersoff Potential can be found in reference [13].

C. Finite Element

1. Model

The Model for the analysis was created using the scripting facility of CATIA (model shown in Fig). Theanalysis and post processing of the model was performed in ABAQUS. The model was meshed using beamelements, and given stiffness parameters of Youngs Modulus(E) of , and a beam radius (r) of . The links areconsidered to be beams, while spring elements are not considered in the current model.

Figure 3. CNT used in the Simulation

2. Buckling Analysis

The buckling analysis of the CNT model using ABAQUS yields the buckling modes as shown below.However, a detailed analysis of a parametrized shell model has to be implemented in order to obtain theelasticity constants which can be used for the ABAQUS model.

III. Results and Discussion

A. Spring Model

On scaling the experimental curve and varying the values of the non dimensional spring constants, a verygood fit of the analytical model to the experimental result is obtained as shown in the figure below.

The dotted line shows the experimental result scaled by a factor of 3, and the thick lines show the equi-librium curves for the spring system. As seen from the curves, there is a very good correlation betweencharacteristics of the spring model and the experimental data. Hence, the behavior of long CNTs (aspect

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Figure 4. Comparison of Spring System with Experiments

ratios¿ 100), can be approximated using the non linear spring model.

B. GULP

The Force Displacement Curves show that there is a rapid decrease in the equilibrium load of the CNT onbuckling, but the load rapidly increases and stabilizes. In this study, only SWCNTsare studied. However,in most of the experiments, the emphasis has been on the study of compression of Multi Walled CNTs(MWCNT ), due to the difficulty of manufacturing a SWCNT . Hence, simulations on MWCNTs wouldgive a better understanding of the nature of buckling and axial stability of the CNT .

1. Force Displacement Curves

Shown in Fig is the Force Displacement curve for a 6-6 CNT having a length of 15A. The aspect ratio for theCNT is The relations to obtain the radius of the CNT can be obtained from the referenceThe energy (E)displacement and force ( F = dE/dx) displacement curves are obtained for the CNT configuration givenabove. We can see that there is a drastic drop in the load at a displacement of , at which the CNT buckles.We can see that after the steep decrease, the axial load increases and then stabilizes for the CNT .

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Figure 5. Energy and Force Displacement Curves

C. FEA

Analysis using ABAQUS gives the mode shapes for the CNT model. However, these results are dependenton the elastic properties assigned to the beam elements in the model. Further, to get results which capturethe physics of compression of the CNT , the joints would need to be modeled with rotational springs, whichwould give a more realistic picture of the buckling of the nanotube. The buckling mode shapes from theanalysis of the CNT are shown in Fig.

Figure 6. First and Second Buckling Modes

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Figure 7. Third and Fourth Buckling Modes

Appendix

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AE 518 Project2 DOF ModelAdd Spring and perform nondimensionNon-linear springs!!!

Potential Energy

Clear��Θ, �x�;urot � Integrate�kr1 �Θ � kr2 �Θ3, �Θ�;utr1 � Integrate�kt1 �x, �x�;�Θ � 2 Θ;

�x �R

2;

��Load displacement���U � L � �L � R� Cos��;��Add another spring��

utr2 �1

2kt2 �U2 �

1

4kt3 �U4;

u � urot � 2 utr1 � utr2;

w � P �U;

��Total potential energy��� � u � w;

Print�"��", ��

��kt1 R2

4� 2 kr1 Θ2 � 4 kr2 Θ4 � P �L � �L � R� Cos�Θ�� �

1

2kt2 �L � �L � R� Cos���2 �

1

4kt3 �L � �L � R� Cos���4

NondimentionΠ � �

4 kr1

r � RL

p � P4 kr1

k1 ��kt1 L2�

32 kr1

k2 ��kt2 L2�

32 kr1

k3 ��kt3 L2�

32 kr1

k22 �kr2

kr1

Π0 �1

2Θ2 � k22 Θ4 � 2 k1 r2 � 4 k2 �1 � �1 � r� Cos�Θ��2 � p �1 � r� Cos�Θ� � p � 2 k3 �1 � �1 � r� Cos�Θ��4;

�u � 1 � �1 � r� Cos�Θ�;��Equlibrium��f1 � Solve�D�Π0, r� 0, p�;f2 � Solve�D�Π0, Θ� 0, p�;pp1 � p �. f1��1��;pp2 � p �. f2��1��;solr � Solve�pp1 pp2, r�

��r �k1 � Cos�Θ� �k1 Θ Csc�Θ� Sec�Θ� � 4 k1 k22 Θ3 Csc�Θ� Sec�Θ� � k12 Sec�Θ�2

2 k1�,

�r �k1 � Cos�Θ� �k1 Θ Csc�Θ� Sec�Θ� � 4 k1 k22 Θ3 Csc�Θ� Sec�Θ� � k12 Sec�Θ�2

2 k1��

r1 � r �. solr��1, 1��;r2 � r �. solr��2, 1��;p1 � pp1 �. r � r1;

p2 � pp1 �. r � r2;

u1 � �u �. r � r1;

u2 � �u �. r � r2;

2 Math_model_v5.nb

plot1 � ParametricPlot�u1 �. k22 � 0.4, k3 � 1, k2 � 1, k1 � 1, p1 �. k22 � 0.4, k3 � 1, k2 � 1, k1 � 1,Θ, 0.001, 1.5, AspectRatio � 1, PlotStyle � Blue, Thick�;

plot2 � ParametricPlot�u2 �. k22 � 0.4, k3 � 1, k2 � 1, k1 � 1,p2 �. k22 � 0.4, k3 � 1, k2 � 1, k1 � 1,Θ, 0.001, 1.5, AspectRatio � 1, PlotStyle � Green, Thick�;

��For Θ�0����u0��u�.Θ�0;��solu0 � Solve���u �. Θ � 0� u0, r�;r0 � r �. solu0��1��;p0 � pp1 �. Θ � 0, r � r0;plot0 � Plot�p0 �. k22 � 0.4, k3 � 1, k2 � 1, k1 � 1, u0, 0, 1, PlotStyle � Red, Thick�;plot00 � ListLinePlot�0, 0, 1.9 � 3.5, 24.5 � 3.5,

1.9 � 3.5, 15 � 3.5, 2.9 � 3.5, 30 � 3.5, PlotStyle � Dashed�;Show�plot0, plot1, plot2, plot00, AspectRatio � 1,

AxesLabel � Displacement, Force, LabelStyle � 16�

0.2 0.4 0.6 0.8 1.0Displacement

5

10

15

20Force

Math_model_v5.nb 3

plot1 � ParametricPlot�u1 �. k22 � 0.5, k3 � 1, k2 � 1, k1 � 1, p1 �. k22 � 0.5, k3 � 1, k2 � 1, k1 � 1,Θ, 0.001, 1.5, AspectRatio � 1, PlotStyle � Blue, Thick�;

plot2 � ParametricPlot�u2 �. k22 � 0.5, k3 � 1, k2 � 1, k1 � 1,p2 �. k22 � 0.5, k3 � 1, k2 � 1, k1 � 1,Θ, 0.001, 1.5, AspectRatio � 1, PlotStyle � Green, Thick�;

��For Θ�0����u0��u�.Θ�0;��solu0 � Solve���u �. Θ � 0� u0, r�;r0 � r �. solu0��1��;p0 � pp1 �. Θ � 0, r � r0;plot0 � Plot�p0 �. k22 � 0.5, k3 � 1, k2 � 1, k1 � 1, u0, 0, 1, PlotStyle � Red, Thick�;plot00 � ListLinePlot�0, 0, 1.9 � 3.5, 24.5 � 3.5,

1.9 � 3.5, 15 � 3.5, 2.9 � 3.5, 30 � 3.5, PlotStyle � Dashed�;Show�plot0, plot1, plot2, AspectRatio � 1, AxesLabel � Displacement, Force, LabelStyle � 16�

0.2 0.4 0.6 0.8 1.0Displacement

5

10

15

20Force

4 Math_model_v5.nb

� CODE FOR CREATING THE CNT MODEL IN CATIA.clear

clc

as � �� CNT COORDINATES COME HERE ��;

fid � fopen �' catia_script _point _final1.txt', ' a'�;

fprintf �fid, ' � s \n � s \n � s \n � s \n � s \n � s \n', ' Language � "VBSCRIPT"', '

Sub CATMain ��', ' Set partDocument1 � CATIA.ActiveDocument ', ...' Set part1 �

partDocument1.Part', ' Set hybridShapeFactory1 � part1.HybridShapeFactory '�;

for df � 1 : size �as, 1�

str1 � �' Set hybridShapePointCoord ' num2str �df�' � hybridShapeFactory1.AddNewPointCoord�' num2str �as �df, 1��', ' num2str �as �df, 2��', ' num2str �as �df, 3��'�'�;

strr1 � �' Set bodies1 � part1.Bodies'�;

strrr1 � �' Set body1 � bodies1.Item �"PartBody"�'�;

str2 � �' body1''.InsertHybridShape hybridShapePointCoord ' num2str �df��;

str3 � �' part1''.InWorkObject � hybridShapePointCoord ' num2str �df��;

str4 � �' part1.Update'�;

fprintf �fid, ' � s \n � s \n � s \n � s \n � s \n � s \n', str1, strr1, strrr1, str2, str3, str4�;� df

� str5 � �''�;

� for fdf � 1 : 5

� dlmwrite �' catia_script.txt', eval ��' str' num2str �fdf���, ' � append '�;� end

end

fprintf �fid, ' � s \n', ' End Sub'�;fclose �fid�

� CODE FOR CREATING POINTS

� CODE FOR CREATING LINES

clear

clc

as � �� CNT COORDINATES COME HERE ��;

count � 1;

for dfx1 � 1 : size �as, 1� � 1for dfx2 � dfx1 � 1 : size �as, 1�if ��as �dfx1, 1� � as �dfx2, 1��^2 �

�as �dfx1, 2� � as �dfx2, 2��^2 � �as �dfx1, 3� � as �dfx2, 3��^2� � 1.43^2pointt �count, :� � �dfx1, dfx2�;count � count � 1;

end

end

end

fid � fopen �' catia_script _CNT _line _final1.txt', ' a'�;fprintf �fid, ' � s \n � s \n � s \n � s \n � s \n � s \n � s \n � s \n �

s \n', ' Language � "VBSCRIPT"', ' Sub CATMain ��', ' Set partDocument1 �

CATIA.ActiveDocument ', ' Set part1 � partDocument1.Part', ...' Set hybridShapeFactory1 �

part1.HybridShapeFactory ', ' Set bodies1 � part1.Bodies', ' Set body1 �

bodies1.Item �"PartBody"�', ...' Set hybridShapes1 � body1.HybridShapes '�;

for df � 1 : size �pointt, 1�

str1 � �' Set hybridShapePointCoord ' num2str �df � 2 � 1�' �hybridShapes1.Item �"Point.' num2str�pointt�df,1�� '"�'�;

str2 � �' Set reference ' num2str �df � 2 � 1�' �part1.CreateReferenceFromObject �hybridShapePointCoord ' num2str �df � 2 � 1�'�'�;

2 CNT.nb

str3 � �' Set hybridShapePointCoord ' num2str �df � 2�' �hybridShapes1.Item �"Point.' num2str�pointt�df,2�� '"�'�;

str4 � �' Set reference ' num2str �df � 2�' �part1.CreateReferenceFromObject �hybridShapePointCoord ' num2str �df � 2�'�'�;

str5 � �' Set hybridShapeLinePtPt ' num2str �df�' � hybridShapeFactory1.AddNewLinePtPt�reference ' num2str �df � 2 � 1�', reference ' num2str �df � 2�'�'�;

str6 � �' body1.InsertHybridShape hybridShapeLinePtPt ' num2str �df��;

str7 � �' part1.InWorkObject � hybridShapeLinePtPt ' num2str �df��;

str8 � �' part1.Update'�;

fprintf �fid, ' � s \n � s \n � s \n � s \n � s \n � s \n � s \n � s \n',

str1, str2, str3, str4, str5, str6, str7, str8�;� df

� str5 � �''�;

� for fdf � 1 : 5

� dlmwrite �' catia_script.txt', eval ��' str' num2str �fdf���, ' � append '�;� end

end

fprintf �fid, ' � s \n', ' End Sub'�;fclose �fid�

CNT.nb 3 4 CNT.nb

���� CODE FOR GULP AUTOMATION ����

�������������������������������

� function CNT_AUTO

dbstop if error

clear

close all

clc

� � � � � INPUTS TO BE GIVEN BY THE USER � � � � �

� input_atoms � dlmread �' input.txt'�;fid_input � fopen �' input.txt', ' r'�;

count � 1;

while�feof �fid_input�linein � fgetl �fid_input�;linein � strrep �linein, ' C', ''�;linein � textscan �linein, ' � f'�;input_atoms �count, :� � linein �1�;count � count � 1;

end

clear count;

fclose �fid_input�;

num_atoms � size �input_atoms, 1�;length_x � max �input_atoms � : , 3�� � min �input_atoms � : , 3��;

� num_atoms � 360;

� length_x � 36.9303;

maxsize � 100;

del_x � length_x � 40;del_x _init � del_x;

� � � � � � � � � END OF INPUTS � � � � � � � � �

flag_optim � 0;

fid � fopen �' nanotube7.in', ' w'�;fprintf �fid, ' � s \n � s \n', ' opti molq conv', ' cell'�;� fprintf �fid, ' � s \n', ' time 50'�;fprintf

�fid, ' � s \n', num2str �maxsize�'' num2str �maxsize�'' num2str �length_x�' 90 � 90 � 90'�;fprintf �fid, ' � s \n', ' cartesian ' num2str �num_atoms��;

fid_input � fopen �' input.txt', ' r'�;for gf � 1 : num_atoms

fprintf �fid, ' � s \n', fgetl �fid_input��;end

fprintf �fid, ' � s \n � s \n', ' library tersoff', ' output arc nanotube ' num2str �length_x��;dos �' a7.bat'�; �� UNCOMMENTpause �10�;fclose �fid�

for auto_runs � 1 : 20

� � � � � FINDING IF OPTIMISATION HAS BEEN ACHIEVED IN PREVIOUS STEP � � � � �

fid_output � fopen �' output.txt'�;count_optim � 0;

length_x1 � length_x

length_x � length_x � del_x;

aline � fgetl �fid_output�;while�feof �fid_output�aline � fgetl �fid_output�;

if�isempty �findstr �aline, ' Optimisation achieved '��count_optim � count_optim � 1;

end

end

if��count_optim � 0�� astr � ' Optimisation achieved '

flag_optim � 1;

else

flag_optim � 0;

end

fclose �fid_output�;� � � � � � � END OF FINDING OPTIMISATION � � � � � � � �

if flag_optim � 1

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

2 CULP.nb

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � STORING THE FILE OUTPUT.TXT IN OUTPUT1.TXT � � � � � � �

if�isempty �dir �' output.txt'��output_fid � fopen �' output.txt', ' r'�;output_fid1 � fopen �' output1.txt', ' w'�;while�feof �output_fid�output_line � fgetl �output_fid�;fprintf �output_fid1, ' � s \n', output_line�;end

fclose �output_fid�;fclose �output_fid1�;end

� � � � � � � STORING THE FILE OUTPUT.TXT IN OUTPUT1.TXT ENDS � � � � � � �

� � � � � � � STORING THE FILE ENERGY.TXT IN ENERGY1.TXT � � � � � � �

if�isempty �dir �' energy.txt'��energy_fid � fopen �' energy.txt', ' r'�;energy_fid1 � fopen �' energy1.txt', ' w'�;while�feof �energy_fid�energy_line � fgetl �energy_fid�;fprintf �energy_fid1, ' � s \n', energy_line�;end

fclose �energy_fid�;fclose �energy_fid1�;end

� � � � � � � STORING THE FILE ENERGY.TXT IN ENERGY1.TXT ENDS � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

fid � fopen �' nanotube7.in', ' w'�;fprintf �fid, ' � s \n � s \n', ' opti molq conv', ' cell'�;� fprintf �fid, ' � s \n', ' time 50'�;fprintf �fid, ' � s \n', ' 100 � 100' num2str �length_x�' 90 � 90 � 90'�;fprintf �fid, ' � s \n', ' fractional ' num2str �num_atoms��;

� � � � � FINDING THE LINES IN THE OUTPUT FILE

WHICH CONTAIN THE FRACTIONAL COORDINATES OF ATOMS � � � � �

fid_output � fopen �' output.txt'�;count_reqd _line � 1;

aline � fgetl �fid_output�;

CULP.nb 3

count_flag � 1;

while count_flag � 1

aline � fgetl �fid_output�;if�isempty �findstr �aline, ' Final fractional coordinates of atoms : '��count_flag � 0;

end

count_reqd _line � count_reqd _line � 1;

end

fclose �fid_output�;� � � � � FINDING THE LINES IN THE OUTPUT FILE WHICH

CONTAIN THE FRACTIONAL COORDINATES OF ATOMS ENDS HERE � � � � �

� � � � � � � READING THE OUTPUT FILE AND WRITING THE REQUIRED DATA INTO THE INPUT FILE � � � � � � �

fid_output � fopen �' output.txt', ' r'�;for hj1 � 1 : count_reqd _line � 5

bline � fgetl �fid_output�;end

for hj2 � 1 : num_atoms

cline � fgetl �fid_output�;dline � textscan �cline, ' � s'�;eline � �dline �1� �2�'' dline �1� �4�'' dline �1� �5�'' dline �1� �6��;fprintf �fid, ' � s \n', eline�;end

� � � � � � � END OF READING THE OUTPUT FILE AND

WRITING THE REQUIRED DATA INTO THE INPUT FILE � � � � � � �

� � � � � � � UPDATE VMD VIEWER FILE � � � � � � �

readarc �num2str �length_x1�, maxsize�;� � � � � � � END OF UPDATE VMD VIEWER FILE � � � � � � �

� � � � � � � STORE ENERGY OF THE SYSTEM IN A SEPARATE FILE � � � � � � � �

� � � � � � � CONSIDERING THE INTERNAL ENERGY TO BE THE LATTICE ENERGY � � � � �

fid_output � fopen �' output.txt'�;count_reqd _line � 1;

aline � fgetl �fid_output�;

count_flag � 1;

count_flag1 � 1;

4 CULP.nb

���� CODE FOR GULP AUTOMATION ����

�������������������������������

� function CNT_AUTO

dbstop if error

clear

close all

clc

� � � � � INPUTS TO BE GIVEN BY THE USER � � � � �

� input_atoms � dlmread �' input.txt'�;fid_input � fopen �' input.txt', ' r'�;

count � 1;

while�feof �fid_input�linein � fgetl �fid_input�;linein � strrep �linein, ' C', ''�;linein � textscan �linein, ' � f'�;input_atoms �count, :� � linein �1�;count � count � 1;

end

clear count;

fclose �fid_input�;

num_atoms � size �input_atoms, 1�;length_x � max �input_atoms � : , 3�� � min �input_atoms � : , 3��;

� num_atoms � 360;

� length_x � 36.9303;

maxsize � 100;

del_x � length_x � 40;del_x _init � del_x;

� � � � � � � � � END OF INPUTS � � � � � � � � �

flag_optim � 0;

fid � fopen �' nanotube7.in', ' w'�;fprintf �fid, ' � s \n � s \n', ' opti molq conv', ' cell'�;� fprintf �fid, ' � s \n', ' time 50'�;fprintf

�fid, ' � s \n', num2str �maxsize�'' num2str �maxsize�'' num2str �length_x�' 90 � 90 � 90'�;fprintf �fid, ' � s \n', ' cartesian ' num2str �num_atoms��;

fid_input � fopen �' input.txt', ' r'�;for gf � 1 : num_atoms

fprintf �fid, ' � s \n', fgetl �fid_input��;end

fprintf �fid, ' � s \n � s \n', ' library tersoff', ' output arc nanotube ' num2str �length_x��;dos �' a7.bat'�; �� UNCOMMENTpause �10�;fclose �fid�

for auto_runs � 1 : 20

� � � � � FINDING IF OPTIMISATION HAS BEEN ACHIEVED IN PREVIOUS STEP � � � � �

fid_output � fopen �' output.txt'�;count_optim � 0;

length_x1 � length_x

length_x � length_x � del_x;

aline � fgetl �fid_output�;while�feof �fid_output�aline � fgetl �fid_output�;

if�isempty �findstr �aline, ' Optimisation achieved '��count_optim � count_optim � 1;

end

end

if��count_optim � 0�� astr � ' Optimisation achieved '

flag_optim � 1;

else

flag_optim � 0;

end

fclose �fid_output�;� � � � � � � END OF FINDING OPTIMISATION � � � � � � � �

if flag_optim � 1

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

2 CULP.nb

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � STORING THE FILE OUTPUT.TXT IN OUTPUT1.TXT � � � � � � �

if�isempty �dir �' output.txt'��output_fid � fopen �' output.txt', ' r'�;output_fid1 � fopen �' output1.txt', ' w'�;while�feof �output_fid�output_line � fgetl �output_fid�;fprintf �output_fid1, ' � s \n', output_line�;end

fclose �output_fid�;fclose �output_fid1�;end

� � � � � � � STORING THE FILE OUTPUT.TXT IN OUTPUT1.TXT ENDS � � � � � � �

� � � � � � � STORING THE FILE ENERGY.TXT IN ENERGY1.TXT � � � � � � �

if�isempty �dir �' energy.txt'��energy_fid � fopen �' energy.txt', ' r'�;energy_fid1 � fopen �' energy1.txt', ' w'�;while�feof �energy_fid�energy_line � fgetl �energy_fid�;fprintf �energy_fid1, ' � s \n', energy_line�;end

fclose �energy_fid�;fclose �energy_fid1�;end

� � � � � � � STORING THE FILE ENERGY.TXT IN ENERGY1.TXT ENDS � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

fid � fopen �' nanotube7.in', ' w'�;fprintf �fid, ' � s \n � s \n', ' opti molq conv', ' cell'�;� fprintf �fid, ' � s \n', ' time 50'�;fprintf �fid, ' � s \n', ' 100 � 100' num2str �length_x�' 90 � 90 � 90'�;fprintf �fid, ' � s \n', ' fractional ' num2str �num_atoms��;

� � � � � FINDING THE LINES IN THE OUTPUT FILE

WHICH CONTAIN THE FRACTIONAL COORDINATES OF ATOMS � � � � �

fid_output � fopen �' output.txt'�;count_reqd _line � 1;

aline � fgetl �fid_output�;

CULP.nb 3

count_flag � 1;

while count_flag � 1

aline � fgetl �fid_output�;if�isempty �findstr �aline, ' Final fractional coordinates of atoms : '��count_flag � 0;

end

count_reqd _line � count_reqd _line � 1;

end

fclose �fid_output�;� � � � � FINDING THE LINES IN THE OUTPUT FILE WHICH

CONTAIN THE FRACTIONAL COORDINATES OF ATOMS ENDS HERE � � � � �

� � � � � � � READING THE OUTPUT FILE AND WRITING THE REQUIRED DATA INTO THE INPUT FILE � � � � � � �

fid_output � fopen �' output.txt', ' r'�;for hj1 � 1 : count_reqd _line � 5

bline � fgetl �fid_output�;end

for hj2 � 1 : num_atoms

cline � fgetl �fid_output�;dline � textscan �cline, ' � s'�;eline � �dline �1� �2�'' dline �1� �4�'' dline �1� �5�'' dline �1� �6��;fprintf �fid, ' � s \n', eline�;end

� � � � � � � END OF READING THE OUTPUT FILE AND

WRITING THE REQUIRED DATA INTO THE INPUT FILE � � � � � � �

� � � � � � � UPDATE VMD VIEWER FILE � � � � � � �

readarc �num2str �length_x1�, maxsize�;� � � � � � � END OF UPDATE VMD VIEWER FILE � � � � � � �

� � � � � � � STORE ENERGY OF THE SYSTEM IN A SEPARATE FILE � � � � � � � �

� � � � � � � CONSIDERING THE INTERNAL ENERGY TO BE THE LATTICE ENERGY � � � � �

fid_output � fopen �' output.txt'�;count_reqd _line � 1;

aline � fgetl �fid_output�;

count_flag � 1;

count_flag1 � 1;

4 CULP.nb

Acknowledgments

We would like to express our gratitude to Professor Anthony M. Waas and Professor Veera Sundararagha-van for their kindness and support during the course of this project.

References

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